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Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander Verdonschot Carleton University 10 August 2015 Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations Pointed: all vertices are incident to a

  1. Flips in Edge-Labelled Pseudo-Triangulations Prosenjit Bose Sander Verdonschot Carleton University 10 August 2015 Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  2. • Pointed: all vertices are incident to a reflex angle ( ) Pseudo-triangulations • All faces are pseudo-triangles Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  3. Pseudo-triangulations • All faces are pseudo-triangles • Pointed: all vertices are incident to a reflex angle ( > π ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  4. Pseudo-triangulations • All faces are pseudo-triangles • Pointed: all vertices are incident to a reflex angle ( > π ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  5. Flips • Remove edge, leaving a pseudo-quadrilateral • Find corners opposite removed edge • Insert connecting geodesic Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  6. • What happens when edges are labelled? Previous work . Theorem (Bereg, 2004) . Any pointed pseudo-triangulation can be transformed into any other with O ( n log n ) flips. . Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  7. Previous work . Theorem (Bereg, 2004) . Any pointed pseudo-triangulation can be transformed into any other with O ( n log n ) flips. . • What happens when edges are labelled? Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  8. • Via canonical form T C • We only need to show T T C Upper bound • Transform T 1 into T 2 Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  9. • We only need to show T T C Upper bound • Transform T 1 into T 2 • Via canonical form T C Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  10. Upper bound • Transform T 1 into T 2 • Via canonical form T C • We only need to show T �→ T C Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  11. Transform into canonical • Ignore labels • Move labels around Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  12. Left-shelling pseudo-triangulation • Add vertices in clockwise order around bottom vertex • Connect to bottom (bottom edge) • Add tangent to convex hull (top edge) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  13. Tools • Sweep : exchange labels on top and bottom pairs • Shuffle : reorder bottom labels Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  14. Algorithm • Identify out-of-place top and bottom labels Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  15. Algorithm • Pair these up ( Shuffle ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  16. Algorithm • Exchange them ( Sweep ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  17. Algorithm • Sort bottom labels ( Shuffle ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  18. Algorithm • Sort bottom labels ( Shuffle ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  19. Algorithm • Move all top labels down ( Sweep ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  20. Algorithm • Move all top labels down ( Sweep ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  21. Algorithm • Sort them ( Shuffle ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  22. Algorithm • Move them back ( Sweep ) Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  23. • How do we shuffle and sweep? Upper bound . Theorem . We can sort the labels of a left-shelling pseudo-triangulation with O ( 1 ) shuffles and sweeps. . Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  24. Upper bound . Theorem . We can sort the labels of a left-shelling pseudo-triangulation with O ( 1 ) shuffles and sweeps. . • How do we shuffle and sweep? Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  25. Sweep • Easy for degree-2 vertices: • Idea: make every vertex degree-2 at some point Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  26. When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Sweep • Shoot a ray from v bottom to the right Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  27. When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Sweep • Sweep it counter-clockwise through the point set Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  28. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  29. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  30. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  31. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  32. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  33. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  34. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  35. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  36. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  37. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  38. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  39. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  40. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  41. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  42. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  43. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  44. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  45. Sweep • When it passes a vertex: • Swap the top and bottom edge, if necessary • Flip the top edge Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  46. Tools • Sweep : exchange labels on top and bottom pairs — O ( n ) • Shuffle : reorder bottom labels Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  47. • Otherwise, flip top edge first • We can do insertion sort! Shuffle • Swap consecutive bottom edges • Easy if third pseudo-triangle is a triangle Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  48. • We can do insertion sort! Shuffle • Swap consecutive bottom edges • Easy if third pseudo-triangle is a triangle • Otherwise, flip top edge first Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

  49. Shuffle • Swap consecutive bottom edges • Easy if third pseudo-triangle is a triangle • Otherwise, flip top edge first • We can do insertion sort! Prosenjit Bose, Sander Verdonschot Flips in Edge-Labelled Pseudo-Triangulations

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